AS4021 Gravitational Dynamics 1 Gravitational Dynamics: An Introduction HongSheng Zhao.

AS4021 Gravitational Dynamics 1

Gravitational Dynamics: An Introduction

HongSheng Zhao


C1.1.1 Our Galaxy and Neighbours

• How structure in universe form/evolve?

• Galaxy Dynamics Link together early universe & future.


Our Neighbours

• M31 (now at 500 kpc) separated from MW a Hubble time ago

• Large Magellanic Cloud has circulated our Galaxy for about 5 times at 50 kpc– argue both neighbours move with a typical

100-200km/s velocity relative to us.


Outer Satellites on weak-g orbits around Milky Way

~ 50 globulars on weak-g (R<150 kpc)~100 globulars on strong-g (R< 10 kpc)

R>10kpc: Magellanic/Sgr/Canis streamsR>50kpc: Draco/Ursa/Sextans/Fornax…


C1.1.2 Milky Way as Gravity Lab

• Sun has circulated the galaxy for 30 times– velocity vector changes direction +/- 200km/s

twice each circle ( R = 8 kpc )– ArgueArgue that the MW is a nano-earth-gravity Lab– Argue that the gravity due to 1010 stars only

within 8 kpc is barely enough. Might need to add Dark Matter.


Sun escapes unless our Galaxy has Dark Matter


C1.1.3 Dynamics as a tool

• Infer additional/dark matter– E.g., Weakly Interacting Massive Particles

• proton mass, but much less interactive

• Suggested by Super-Symmetry, but undetected

– A $billion$ industry to find them. • What if they don’t exist?


…

• Test the law of gravity: – valid in nano-gravity regime?– Uncertain outside solar system:

• GM/r2 or cst/r ?


Outer solar system

• The Pioneer experiences an anomalous non-Keplerian acceleration of 10-8 cm s-2

• What is the expected acceleration at 10 AU?

• What could cause the anomaly?


Gravitational Dynamics can be applied to:

• Two body systems:binary stars

• Planetary Systems, Solar system

• Stellar Clusters:open & globular

• Galactic Structure:nuclei/bulge/disk/halo

• Clusters of Galaxies

• The universe:large scale structure


Topics• Phase Space Fluid f(x,v)

– Eqn of motion– Poisson’s equation

• Stellar Orbits– Integrals of motion (E,J)– Jeans Theorem

• Spherical Equilibrium– Virial Theorem– Jeans Equation

• Interacting Systems– TidesSatellitesStreams– Relaxationcollisions

• MOND


C2.1 How to model motions of 1010stars in a galaxy?

• Direct N-body approach (as in simulations)– At time t particles have (mi,xi,yi,zi,vxi,vyi,vzi),

i=1,2,...,N (feasible for N<<106 ).

• Statistical or fluid approach (N very large)– At time t particles have a spatial density

distribution n(x,y,z)*m, e.g., uniform, – at each point have a velocity distribution

G(vx,vy,vz), e.g., a 3D Gaussian.


C2.2 N-body Potential and Force

• In N-body system with mass m1…mN, the gravitational acceleration g(r) and potential φ(r) at position r is given by:

r12

Ri

r mi ∑

∑

=

=

−⋅⋅

−==Φ

∇−=−

⋅⋅⋅−==

N

i i

i

N

ir

i

i

Rr

mmGrm

mRr

rmmGrgmF

1

12

12

)(

ˆ)(

vv

rvv

rr

φ

φ


Example: Force field of two-body system in Cartesian coordinates

0?force is positionsat what lines. fieldsketch

?)()(

),,()(),,()(

?),,(

contours potential equalsketch ion,configurat Sketch the

,),0,0( where,)(

222

2

1

=

=++=

∂∂

−∂∂

−∂∂

−=−∇==

=

=∗−=−⋅

−= ∑=

zyx

zyx

iii i

i

gggrg

zyxrgggrg

zyx

mmaiRRrmG

r

vr

vvr

vvv

vo

φφφφ

φ

φ


C2.3 A fluid element: Potential & Gravity

• For large N or a continuous fluid, the gravity dg and potential dφ due to a small mass element dM is calculated by replacing mi with dM:

r12

Rr

dM

d3RRr

dMGd vv−

⋅−=φ

212ˆ

iRr

rdMGgd vvr

−

⋅⋅−=


Lec 2 (Friday, 10 Feb): Why Potential φ(r) ?

• Potential per unit mass φ(r) is scalar,– function of r only,– Related to but easier to work with than force

(vector, 3 components) – Simply relates to orbital energy E= φ(r) + ½ v2


C2.4 Poisson’s Equation

• PE relates the potential to the density of matter generating the potential by:

• [BT2.1]

)(4 rGg ρπφ =⋅∇−=∇⋅∇rr


C2.5 Eq. of Motion in N-body

• Newton’s law: a point mass m at position r moving with a velocity dr/dt with Potential energy Φ(r) =mφ(r) experiences a Force F=mg , accelerates with following Eq. of Motion:

m

r

m

F

dt

trd

dt

d r )()( Φ∇−==⎥⎦

⎤⎢⎣

⎡ rrvv


Example 1: trajectories when G=0

• Solve Poisson’s Eq. with G=0 – F=0, Φ(r)=cst,

• Solve EoM for particle i initially at (Xi,0, V0,i)– dVi/dt = Fi/mi = 0 Vi = cst = V0,i – dXi/dt = Vi = Vi,0 Xi(t) = Vi,0 t + Xi,0, – where X, V are vectors,

straight line trajectories • E.g., photons in universe go straight

– occasionally deflected by electrons,– Or bent by gravitational lenses


What have we learned?

• Implications on gravity law and DM.

• Poisson’s eq. and how to calculate gravity

• Equation of motion


How N-body system evolves

• Start with initial positions and velocities of all N particles.

• Calculate the mutual gravity on each particle– Update velocity of each particle for a small time step dt

with EoM

– Update position of each particle for a small time step dt

• Repeat previous for next time step. N-body system fully described


C2.6 Phase Space of Galactic Skiers • Nskiers identical particles moving in a small bundle

in phase space (Vol =Δx Δ v),

• phase space deforms but maintains its area.

• Gap widens between faster & slower skiers– but the phase volume & No. of skiers are constants.

+x front

vx

x

+Vx Fast


“Liouvilles Theorem on the piste”

• Phase space density of a group of skiers is const. f = m Nskiers / Δx Δvx = const

Where m is mass of each skier,

[ BT4.1]


C2.7 density of phase space fluid: Analogy with air molecules

• air with uniform density n=1023 cm-3

Gaussian velocity rms velocity σ =0.3km/s in x,y,z directions:

• Estimate f(0,0,0,0,0,0) in pc-3 (km/s)-3

3

2

222

o

)2(

2expnm

v)f(x,σπ

σ ⎟⎟⎠

⎞⎜⎜⎝

⎛ ++−×

=

xyx vvv


Lec 3 (Valentine Tuesday)C2.8 Phase Space Distribution Function (DF)

PHASE SPACE DENSITY: No. of sun-like stars per unit volume per velocity volume f(x,v)

sun sun3 3

sun3 1 3

m number of suns mf(x,v)

space volume velocity volume

1 m

pc (100kms )

dN

dx dv

−

× ×= =

××

×:


C2.9 add up stars: integrate over phase space

• star mass density: integrate velocity volume

• The total mass : integrate over phase space

xdvdvxfxdxM total

vvvv 333 ),()( ∫∫ == ρ


• define spatial density of stars n(x)

• and the mean stellar velocity v(x)

• E.g., Conservation of flux (without proof)( ) ( ) ( )

3

3

2 31

1 2 3

flux in i-direction

0

i i

n fd v

nv fv d v

nv nv nvn

t x x x

∂ ∂ ∂∂

∂ ∂ ∂ ∂

≡

≡ =

+ + + =

∫

∫


C3.0 Star clusters differ from air:

• Stars collide far less frequently– size of stars<<distance between them– Velocity distribution not isotropic

• Inhomogeneous density ρ(r) in a Grav. Potential φ(r)


Example 2: A 4-body problem

• Four point masses with G m = 1 at rest (x,y,z)=(0,1,0),(0,-1,0),(-1,0,0),(1,0,0). Show the initial total energy

Einit = 4 * ( ½ + 2-1/2 + 2-1/2) /2 = 3.8

• Integrate EoM by brutal force for one time step =1 to find the positions/velocities at time t=1.– Use V=V0 + g t = g = (u, u, 0) ; u = 21/2/4 + 21/2/4 + ¼ = 0.95

– Use x= x0 + V0 t = x0 = (0, 1, 0).

• How much does the new total energy differ from initial? E - Einit = ½ (u2 +u2) * 4 = 2 u2 = 1.8


Often-made Mistakes

• Specific energy or specific force confused with the usual energy or force

• Double-counting potential energy between any pair of mass elements, kinetic energy with v2

• Velocity vector V confused with speed, • 1/|r| confused with 1/|x|+1/|y|+1/|z|



Potential to Gravity

Potential to density

Density to potential

Motion to gravity

g φ=−∇

∫ ′−

′−=

rr

rdGr vv

vv 3

)(ρ

φ

/g dv dt=


Concepts

• Phase space density– incompressible– Dimension Mass/[ Length3 Velocity3 ]

– Show a pair of non-relativistic Fermionic particle occupy minimal phase space (x*v)3 > (h/m)3 , hence has a maximum phase density =2m (h/m)-3


Where are we heading to?Lec 4, Friday 17 Feb

• potential and eqs. of motion – in general geometry– Axisymmetric– spherical


Link phase space quantities

r

J(r,v)

Ek(v)

(r)

Vt

E(r,v)dθ/dt

vr


C 3.1: Laplacian in various coordinates

2

2

2222

22

2

2

2

2

22

2

2

2

2

2

22

sin

1sin

sin

11

:Spherical

11

:lCylindrica

:Cartesians

φθθθ

θθ

φ

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

=∇

∂∂

+∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

=∇

∂∂

+∂∂

+∂∂

=∇

rrrr

rr

zRRR

RR

zyx


Example 3: Energy is conserved in STATIC potential

• The orbital energy of a star is given by:

),(2

1 2 trvEvφ+=

0dE dv dr

vdt dt dt t t

φ φφ ∂ ∂= + ∇ + = +

∂ ∂

v vv

0 since

and

φ−∇=dt

vdv

vdt

rd vv=

0 for static potential.

So orbital Energy is Conserved dE/dt=0 only in “time-independent” potential.


Example 4: Static Axisymmetric density Static Axisymmetric potential

• We employ a cylindrical coordinate system (R,,z) e.g., centred on the galaxy and align the z axis with the galaxy axis of symmetry.

• Here the potential is of the form (R,z).• Density and Potential are Static and Axisymmetric

– independent of time and azimuthal angle

zg

Rg

zRR

RR

GzRzR

zr ∂∂

−=∂∂

−=

⎥⎦

⎤⎢⎣

⎡

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

=⇒

φφ

φφπ

ρφ 2

2

41

),(),(


C3.2: Orbits in an axisymmetric potential

• Let the potential which we assume to be symmetric about the plane z=0, be (R,z).

• The general equation of motion of the star is

• Eqs. of motion in cylindrical coordinates

),(2

2

zRdt

rd φ−∇=v

Eq. of Motion


Conservation of angular momentum z-component Jz if axisymmetric

• The component of angular momentum about the z-axis is conserved.

• If (R,z) has no dependence on θ then the azimuthal angular momentum is conserved– or because z-component of the torque rF=0. (Show it)

2 2( ) 0Z

d dJ R Jz R

dt dtθ θ= ⇒ = =& &


C4.1: Spherical Static System

• Density, potential function of radius |r| only

• Conservation of – energy E, – angular momentum J (all 3-components)– Argue that a star moves orbit which confined to

a plane perpendicular to J vector.


C 4.1.0: Spherical Cow Theorem

• Most astronomical objects can be approximated as spherical.

• Anyway non-spherical systems are too difficult to model, almost all models are spherical.


Globular: A nearly spherical static system


C4.2: From Spherical Density to Mass

M(r)

M(r+dr)

∫ ⎟⎠

⎞⎜⎝

⎛=

=⎟⎠⎞

⎜⎝⎛

=

=⎟⎠

⎞⎜⎝

⎛=

+=+

3

23

23

3

4)(

434

)(

(r)43

4(r)ddM

dMM(R)dr)M(R

rdRM

drr

dM

rd

dMr

drrr

πρ

ππ

ρ

ρππρ


C4.3: Theorems on Spherical Systems

• NEWTONS 1st THEOREM:A body that is inside a spherical shell of matter experiences no net gravitational force from that shell

• NEWTONS 2nd THEOREM:The gravitational force on a body that lies outside a closed spherical shell of matter is the same as it would be if all the matter were concentrated at its centre. [BT 2.1]


C4.4: Poisson’s eq. in Spherical systems

• Poisson’s eq. in a spherical potential with no θ or Φ dependences is:

• BT2.1.2

)(41 2

22 rG

rr

rrρπ =⎟

⎠

⎞⎜⎝

⎛∂∂

∂∂

=∇


Example 5: Interpretation of Poissons Equation

• Consider a spherical distribution of mass of density ρ(r).

r

g

drrr

drr

rGM

drrg

r

rGMg

r

r

r

)(4Enclosed Mass

)(

rat 0 is andat0 since)(

)(

2

2

2

ρπ

φφ

∫

∫

∫

∞

∞

∞

=

−=

<∞==

−=


• Take d/dr and multiply r2

• Take d/dr and divide r2

( )∫==−= drrrGrGMgrdr

dr )(4)( 222 ρπ

φ

( ) ( )

ρπφ

ρπ

Gg

rGGMrr

grrrr

rrr

4.

)(4111

2

22

22

2

=∇−=∇→

=∂

∂=−

∂

∂=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂

v


C4.5: Escape Velocity• ESCAPE VELOCITY= velocity required in

order for an object to escape from a gravitational potential well and arrive at with zero KE.

21( ) ( )

2

( ) 2 ( ) 2 ( )

esc

esc

r v

v r r

φ φ

φ φ

= ∞ −

→ = ∞ −

=0 often


Example 6: Plummer Model for star cluster

• A spherically symmetric potential of the form:

• Show corresponding to a density (use Poisson’s eq): 2

5

2

2

31

4

3−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ar

aMπ

ρ

e.g., for a globular cluster a=1pc, M=105 Sun Mass show Vesc(0)=30km/s



• Conditions for conservation of orbital energy, angular momentum of a test particle

• Meaning of escape velocity

• How Poisson’s equation simplifies in cylindrical and spherical symmetries


Lec 5, (Tue 21 Feb)Links of quantities in spheres

g(r)

(r) ρ(r)

M(r)


A worked-out example 7: Hernquist Potential for stars in a galaxy

• E.g., a=1000pc, M0=1010 solar, show central escape velocity Vesc(0)=300km/s,

• Show M0 has the meaning of total mass

– Potential at large r is like that of a point mass M0

– Integrate the density from r=0 to inifnity also gives M0

0*

1 2

0* 3

( ) , use Poisson eq. show

( ) 12

GMr

a r

M r rr

a a a

φ

ρπ

− −

=−+

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠


Potential of globular clusters and galaxies looks like this:

r

a

GM o−

-1

(0) (finite well at centre)

( ) r (Kepler for large r)

is the minimum of potential with escape velocity

2(0) o

esc

const

r

Centre

GMv

a

φφ

=−

∝→

=


C4.6: Circular Velocity

• CIRCULAR VELOCITY= the speed of a test particle in a circular orbit at radius r.

G

rvrM

r

rGM

r

vg

cir

cir

2

2

2

)(

)(

=⇒

=∇== φ

For a point mass M: Show in a uniform density sphere

r

GMrvc =)( ρπρπ 3

3

4M(r)since

3

4)( rr

Grvc ==



• How to apply Poisson’s eq.

• How to relate – Vesc with potential and – Vcir with gravity

• The meanings of – the potential at very large radius,– The enclosed mass


Lec 6, (Fri, 24 Feb)Links of quantities in spheres

g(r)

(r) ρ(r)

vesc

M(r)

Vcir


C4.7: Motions in spherical potential [BT3.1]

If spherical

0

rg r

gθ

φ

φθ

∂=−

∂∂

=− =∂

φ−∇==

=

gvdtddtd

motion ofEquation

v

vx

0

00

)(

)(

gravity no If

vv

xvx

=+=

ttt 2

Conserved if spherical static

1E ( )

2ˆt

v r

L J rv n

φ= +

= = ⊗ = ⋅x v


Another Proof: Angular Momentum is Conserved if spherical

•

vrLvvv

×=( )

0d r vdL dr dv

v r r gdt dt dt dt

×= = × + × = + ×

v v v v vrv v v

Since 0=∂∂θφ

then the spherical force g is in the r direction, no torque

both cross products on the RHS = 0.

So Angular Momentum L is Conserved

0dL

dt=

v


C4.8: Star moves in a plane (r,θ) perpendicular to L

• Direction of angular momentum

• equations of motion are– radial acceleration:– tangential acceleration:

2

2

2

( )

2 0

constant

r r L

r r g r

r r

r L

θθ θθ

× =

− =

+ =

= =

&

&&&& &&&&


C4.8.1: Orbits in Spherical Potentials

• The motion of a star in a centrally directed field of force is greatly simplified by the familiar law of conservation (WHY?) of angular momentum.

unit time

sweptarea2r

const

2 ==

=×=

dtddtrd

rL

θ

vvv

Keplers 3rd law


C 4.9: Radial part of motion• Energy Conservation (WHY?)

2

2

2

22

2

1

2)(

2

1

2

1)(

⎟⎠

⎞⎜⎝

⎛++=

⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛+=

dtrd

rL

r

dtd

rdtrd

rE

v

v

φ

θφ

eff

)(22 rEdt

dreffφ−±=


C5.0: Orbit in the z=0 plane of a disk potential (R,z).

• Energy/angular momentum of star (per unit mass)

• orbit bound within

( )2212

221

2 2

21eff2

eff

( ,0)

[ ( ,0)] 2

(R,0)

( ,0)

z

E R R R

JR R

R

R

E R

θ⎡ ⎤= + + Φ⎢ ⎥⎣ ⎦

⎡ ⎤= + + Φ⎣ ⎦

⎡ ⎤= + Φ⎣ ⎦

⇒ ≥ Φ

&&

&

&


C5.1: Radial Oscillation• An orbit is bound between two radii: a loop• Lower energy E means thinner loop (nearly

circular closed) orbit

R

eff

E

2

2

2R

J z

Rcir


C5.2: Eq of Motion for planar orbits

• EoM:

eff

eff

; z=0

If circular orbit R=cst, 0 => 0 at c

RR

z R RR

∂∂

∂∂

Φ=−

Φ= = =

&&


Apocenter

Pericenter


Apocenter

Peri

Apo


C5.3: Apocenter and pericenter

• No radial motion at these turn-around radii– dr/dt =Vr =0 at apo and peri

• Hence• Jz = R Vt

= RaVa = RpVp

• E = ½ (Vr2 + Vt

2 ) + Φ (R,0) = ½ Vr2 + Φeff (R,0)

= ½ Va2 + Φ (Ra,0)

= ½ Vp2 + Φ (Rp,0)


Lec7: Orbit in axisymmetric disk potential (R,z).

• Energy/angular momentum of star (per unit mass)

• orbit bound within

( )22 212

22 21

2 2

2 21eff2

eff

2

(R,z)

( , )

z

E R R z

JR z

R

R z

E R z

θ⎡ ⎤= + + + Φ⎢ ⎥⎣ ⎦

⎡ ⎤= + + + Φ⎣ ⎦

⎡ ⎤= + + Φ⎣ ⎦

⇒ ≥ Φ

&& &

& &

& &


C5.4 EoM for nearly circular orbits

• EoM:

• Taylor expand

– x=R-Rg

eff eff

eff eff

2 22 2eff eff1 1

eff 2 22 2

( ,0) ( ,0)

2 2 2 2

;

0 at , 0

/ 2 / 2 ...

g g

g

R R

R zR z

R R zR z

x zR z

x z

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

κ ν

Φ Φ=− =−

Φ Φ= = = =

⎛ ⎞ ⎛ ⎞Φ ΦΦ = + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

= + +

&& &&

K


Sun’s Vertical and radial epicycles

• harmonic oscillator +/-10pc every 108 yr – epicyclic frequency :– vertical frequency :

2 2 R , and

R z zκ ν=− =−&& &&

κ

ν


Lec 8, C7.0: Stars are not enough: add Dark Matter in galaxies [BT10.4]

NGC 3198 (Begeman 1987)


Bekenstein & Milgrom (1984) Bekenstein (2004), Zhao & Famaey (2006)

• Modify gravity g, – Analogy to E-field in medium of varying Dielectric

• Gradient of Conservative potential

*

0 0

0

0 0

g( )

4 G

G(g/a ) = (1+ a /g) G

~ G if g= | |

~ G a / > G if g <a

r

a

g

ρπ

φ

⎛ ⎞−∇• =⎜ ⎟⎝ ⎠

∇ >

Not Standard Belief!


Researches of Read & Moore (2005)

MOND and DM similar in potential, rotation curves and orbits... Personal view!


Explained: Fall/Rise/wiggles in Ellip/Spiral/Dwarf galaxies

Views of Milgrom & Sanders, Sanders & McGaugh



• Orbits in a spherical potential or in the mid-plane of a disk potential

• How to relate Pericentre, Apocentre through energy and angular momentum conservation.

• Rotation curves of galaxies– Need for Dark Matter or a boosted gravity


Tutorial: Singular Isothermal Sphere

• Has Potential Beyond ro:

• And Inside r<r0

• Prove that the potential AND gravity is continuous at r=ro if

• Prove density drops sharply to 0 beyond r0, and inside r0

• Integrate density to prove total mass=M0

• What is circular and escape velocities at r=r0?

• Draw diagrams of M(r), Vesc(r), Vcir(r), |Phi(r)|, rho(r), |g(r)| vs. r (assume V0=200km/s, r0=100kpc).

r

GMr 0)( −=φ

oor

rvr φφ += ln)( 20

20000 / vrGM −=−=φ

2

20

4)(

Gr

Vr

πρ =


Another Singular Isothermal Sphere• Consider a potential Φ(r)=V0

2ln(r). • Use Jeans eq. to show the velocity dispersion σ (assume isotropic) is

constant V02/n for a spherical tracer population of density A*r-n ; Show

we required constants A = V02/(4*Pi*G). and n=2 in order for the tracer

to become a self-gravitating population. Justify why this model is called Singular Isothermal Sphere.

• Show stars with a phase space density f(E)= exp(-E/σ2) inside this potential well will have no net motion <V>=0, and a constant rms velocity σ in all directions.

• Consider a black hole of mass m on a rosette orbit bound between pericenter r0 and apocenter 2r0 . Suppose the black hole decays its orbit due to dynamical friction to a circular orbit r0/2 after time t0. How much orbital energy and angular momentum have been dissipated? By what percentage has the tidal radius of the BH reduced? How long would the orbital decay take for a smaller black hole of mass m/2 in a small galaxy of potential Φ(r)=0.25V0

2ln(r). ? Argue it would take less time to decay from r0 to r0 /2 then from r0/2 to 0.


Lec 9: C8.0: Incompressible df/dt=0• Nstar identical particles moving in a small

bundle in phase space (Vol=Δx Δ p), • phase space deforms but maintains its area.

– Likewise for y-py and z-pz.

Phase space density f=Nstars/Δx Δ p ~ const

THEOREM'LIOUVILLES',0,0 ==λλ d

dNstard

dVol

px

x

px

x


C8.1: Stars flow in phase-space[BT4.1]

• Flow of points in phase space ~

stars moving along their orbits.

• phase space coords: ),(),(

),...,,(),( 621

Φ−∇==≡≡

vvxwwwwwvx

&&&


C8.2 Collisionless Boltzmann Equation

• Collisionless df/dt=0:

• Vector form

6

1

3

1

d ( , , ) ( , ) 0

dt

0

0

ii i i i

f x v t w f w tt w

f f fv

t x x v

f fv f

t v

αα α

∂ ∂∂ ∂

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

∂ ∂∂ ∂

=

=

⎛ ⎞= + =⎜ ⎟

⎝ ⎠

⎡ ⎤Φ+ − =⎢ ⎥

⎣ ⎦

+ ⋅∇ −∇Φ⋅ =

∑

∑

&


C8.3 DF & its 0th ,1st , 2nd moments

3 3

3

3 3 3

x

2 2 2 2x y z

( , )( )

( , )

where A( , ) 1, V , ...

e.g., verify V V V V

d Af x v dA

d

d dM d f x v d

ρ

ρ

=

= =

=

= + +

∫∫∫

∫∫∫x y

x vx

x

x x v

x v ,V V


• E.g: rms speed of air particles in a box dx3 :2 2 2

o 2

3

m n exp2

f(x,v)( 2 )

x y zv v v

σ

πσ

⎛ ⎞+ +× −⎜ ⎟⎜ ⎟

⎝ ⎠=


C8.4: CBE Moment/Jeans Equations [BT4.2]

• Phase space incompressible

df(w,t)/dt=0, where w=[x,v]: CBE

• taking moments U=1, vj, vjvk by integrating over all possible velocities

6 3

1 1

3 3 3

0

0

ii i i i

ii i i

f f f f fw v

t w t x x v

f f fUd v v Ud v Ud vt x x v

αα α

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= =

⎡ ⎤Φ+ = + − =⎢ ⎥

⎣ ⎦

Φ+ − =

∑ ∑

∫ ∫ ∫

&


C8.5: 0th moment (continuity) eq.

• define spatial density of stars n(x)

• and the mean stellar velocity v(x)

• then the zeroth moment equation becomes( )

3

3

1,2,3

1

0

i i

i

i i

n fd v

v fv d vn

nvn

t x

∂∂

∂ ∂=

≡

≡

+ =

∫

∫

∑


C8.6: 2nd moment Equation

similar to the Euler equation for a fluid flow:– last term of RHS represents pressure force

( ) ( )

( )

2

1,2,3 1,2,3

2

( )( )

1

j ijji

i ii j i

i jij i i j j i j

v nvv

t x x n x

v v v v VV V V

vv v P

t

∂ ∂ σ∂ ∂

∂ ∂ ∂ ∂

σ

∂

∂ ρ

= =

Φ→ + = − −

= − − = −

+ ⋅∇ = −∇Φ − ∇

∑ ∑

( ) Pnx iji

−∇⇔− 2 σ∂

∂


Lec 10, C8.7: Meaning of pressure in star system

• What prevents a non-rotating star cluster collapse into a BH? – No systematic motion as in Milky Way disk.– But random orbital angular momentum of stars!


C8.8: Anisotropic Stress Tensor • describes a pressure which is

– perhaps stronger in some directions than other

• the tensor is symmetric, can be diagonalized– velocity ellipsoid with semi-major axes given

by

σ ij2 = σ ii

2δ ij

σ11 ,σ 22 ,σ 33

2ij ijP nσ=


C8.9: Prove Tensor Virial Theorem (BT4.3)

• Many forms of Viral theorem, E.g.

2 2 2 2

etc

.

Scaler Virial Theorem

2 0

x x x

x y y

x y z

v v x

v v x

v v v v

r

drdr

K W

φφ

φφ

< >=< ∂ >< >=< ∂ >

< >=< > + < > + < >

=< ∇ >

=< >

⇒ + =

v

2 ij ij

vv r

K W

φ⇒< >=< ∇ >=−

vv v


C9.0: Jeans theorem [BT4.4]

• For most stellar systems the DF depends on (x,v) through generally 1,2 or 3 integrals of motion (conserved quantities),

• Ii(x,v), i=1..3 f(x,v) = f(I1(x,v), I2(x,v), I3(x,v))

• E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component

)ˆ||,||,(),( zLLEfvxf ⋅=vv


C9.1: An anisotropic incompressible spherical fluid, e.g, f(E,L) =exp(-E/σ0

2)L2β [BT4.4.4]

• Verify <Vr2> = σ0

2, <Vt2>=2(1-β) σ0

2

• Verify <Vr> = 0

• Verify CBE is satisfied along the orbit or flow:

( , ) ( , ) ( , )0 0

df E L f E L dE f E L dL

dt E dt L dt

∂ ∂= + = +

∂ ∂

0 for static potential, 0 for spherical potential

So f(E,L) indeed constant along orbit or flow


C9.2: Apply JE & PE to measure Dark Matter [BT4.2.1d]

• A bright sub-component of observed density n*(r) and anisotropic velocity dispersions

<Vt2>= 2(1-β)<Vr

2>

• in spherical potential φ(r) from total (+dark) matter density ρ(r)

( )2 2

2

2

02

21JE: - *

*

( )4PE:

t rr

r

v vd dn v

n dr r dr

G r r dr d

r dr

ρ π

− Φ+ =

Φ=

∫

[


C9.3: Measure total Matter density ρr)

Assume anistropic parameter beta, substitute JE for stars of density n* into PE, get

• all quantities on the LHS are, in principle, determinable from observations.

( )222

2

*2 - ( ) ( )

4 *

rr

d n vvd rr

r dr G r n dr

βρ

⎡ ⎤⎢ ⎥+ =⎢ ⎥π⎢ ⎥⎣ ⎦


C9.4: Spherical Isotropic f(E) Equilibriums [BT4.4.3]

• ISOTROPIC β=0:The distribution function f(E) only depends on |V| the modulus of the velocity, same in all velocity directions.

Note:the tangential direction has θ and components


C9.5: subcomponents

• Non-SELF-GRAVITATING: There are additional gravitating matter

• The matter density that creates the potential is NOT equal to the density of stars. – e.g., stars orbiting a black hole is non-self-

gravitating.


C9.6: subcomponents add upto the total gravitational mass

1 2

1 2

Phase density of stars plus dark matter

density of stars plus dark matter

Total density shared total potential

A B A B

f f f

ρ ρ ρ

ρ

+ = +

= +

= +

==>Φ



• Meaning of anistrpic pressure and dispertion.

• Usage of Jeans theorem [phase space]

• Usage of Jeans eq. (dark matter)

• Link among quantities in sphere.


C10.0: Galactic disk mass density from vertical equilibrium

• Use JE and PE in cylindrical coordinates.

• drop terms 1/R or d/dR ( d/dz terms dominates).– |z|< 1kpc < R~8kpc near Sun

• Combine vertical hydrostatic eq. and gravity eq. :

• RHS are observables

• => JE/PE useful in weighing the galaxy disk.

( )2 ** zn v

n z z

∂ ∂

∂ ∂

Φ≈ −

2

24 G

z

∂ρ∂Φ

π ≈

( )z=1kpc

2

1z=-1kpc

total density in a column 1kpc high

2= dz *

4 * zz kpc

n vGn z

∂ρ

∂ =

⇒

⎡ ⎤= ⎢ ⎥π⎣ ⎦∫


A toy galaxy

2 2 2 2 2 2 2 1/ 20 0 ( , ) 0.5 ln( 2 ) (1 ( ) /1 ) ,

0 100 / . Argue 1st & 2nd terms of above

galaxy potential resemble dark halo and stars respectively.

R z v R z v R z kpc

v km s

φ −= + − + +=

Calculate the circular velocity and dark halo density

on equator (R,z) (1kpc,0)

Estimate the total mass of stars and dark matter inside 10kpc.

Estimate the star column density inside |z|<0.1kpc, R=1kpc.

=

If you like challenge…

Learn to analyse


Helpful Math/Approximations(To be shown at AS4021 exam)

• Convenient Units

• Gravitational Constant

• Laplacian operator in various coordinates

• Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube

3dv3dx),(dM

)(spherical 2sin2r

2

sin2r

)(sin

2r

)2(

al)(cylindric 22-R2)(1-R

ar)(rectangul 222

1-sun

M2(km/s) kpc6104

1-sun

M2(km/s) pc3104

Gyr1

kpc1

1Myr

1pc 1km/s

vxf

rr

r

zRR

R

zyx

G

G

=

∂+

∂∂+

∂∂=

∂+∂+∂∂=

∂+∂+∂=∇•∇

−×≈

−×≈

≈≈

θ

φ

θθθθ

φ


Thinking of a globular cluster …


C10.1: Link phase space quantities

r

J(r,v)

K(v)

(r)

Vt

E(r,v)dθ/dt

vr


C10.2: Link quantities in spheres

g(r)

(r) ρ(r)vesc

2(r)

M(r)Vcir2 (r)

σr2(r)

σt2(r)

f(E,L)

AS4021 Gravitational Dynamics 1 Gravitational Dynamics: An Introduction HongSheng Zhao.

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Transcript of AS4021 Gravitational Dynamics 1 Gravitational Dynamics: An Introduction HongSheng Zhao.